Primary resonance suppression in spur gear systems using hybrid proportional and fractional-order derivative displacement feedback control

This paper investigated primary resonance suppression in nonlinear spur gear systems using a hybrid proportional and fractional-order derivative displacement feedback (P-FDDF) controller. The dynamic model of the system was established through a second-order non-autonomous differential equation incorporating time-varying meshing stiffness, backlash, and external excitations. The amplitude-frequency response equation of primary resonance was derived via the multiple scale method, while Lyapunov stability theory was employed to analyze the stability of steady-state solutions. Numerical analyses examined the effects of meshing damping, load fluctuations, meshing stiffness variations, and control parameters on resonance characteristics. Time history responses and phase diagrams demonstrated that the P-FDDF strategy achieves simultaneous resonant amplitude suppression and frequency tuning. The fractional-order component's frequency-weighting and memory properties enhance adaptability to complex nonlinear dynamics induced by time-varying meshing stiffness and backlash, establishing the P-FDDF as a reliable solution for gear system vibration control.